Geometrie der Quanten-Hall-Zustände für große Teilchenzahlen
Zusammenfassung der Projektergebnisse
To summarize, the main results of this project is an improved understanding of various questions for the quantum Hall states on Riemann surfaces. We constructed the basis of β g degenerate Laughlin states on Riemann surfaces (Wen-Niu’s topological degeneracy). We defined the Pfaffian states on a sphere with arbitrary metric on inhomogeneous magnetic field and computed the leading large N corrections to the expectation value of the density and the structure factor. Several works related to this project are still ongoing. With D. Zvonkine (Paris-Versaille) we hope to finally settle the topological degeneracy of Laughlin states (it remains to prove that the β g states is the full basis), and address the conjectured projective flatness of the corresponding bundles on the moduli spaces. With X. Ma and G. Marinescu we computed the Chern classes of the one-particle states on Riemann surfaces and found out that projective flatness in general does not hold for these bundles, which is a surprising and potentially interesting results, since it would imply the existence of higher Chern classes, which are independent (not powers of) of the first Chern class. With N. Nemkov we studied the Pfaffian states on the torus and demonstrated their projective flatness. As the outcome of the project we ”expected to uncover new exciting connections of the QH states with various aspects of modern geometry and physics, with potential interest to the matha ematical physics, Köhler geometry, Bergman kernel, random matrix and QHE communities.” In July 2019 and international conference ”New directions in Mathematics of Coulomb gas and quantum Hall effect” was held in Mittag-Leffler Institute, Stockholm. Basically all the topics raised in this project were discussed and I am happy to report that interest of the mathematical physics community to the questions addressed in this project is indeed strong and growing.
Projektbezogene Publikationen (Auswahl)
- Geometric responses of the Pfaffian state, Phys. Rev. B 99 (2019) 205158
V. Dwivedi and S. Klevtsov
(Siehe online unter https://doi.org/10.1103/physrevb.99.205158) - Laughlin states on higher genus Riemann surfaces, Commun. Math. Phys. 367 (2019) 837–871
S. Klevtsov
(Siehe online unter https://doi.org/10.1007/s00220-019-03318-6)
